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Peirce’s Existential Graphs. Bram van Heuveln Minds and Machines Lab, RPI Summer 2001. Today’s Topics. Alpha symbolization rules of inference some theory Java implementation. Alpha Sheet of Assertion.
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Peirce’s Existential Graphs Bram van Heuveln Minds and Machines Lab, RPI Summer 2001
Today’s Topics • Alpha • symbolization • rules of inference • some theory • Java implementation
AlphaSheet of Assertion • To assert some statement in EG, you put the symbolization of that statement on a sheet of paper, called the ‘Sheet of Assertion’ (SA). The location of the statement on the SA does not matter, i.e: states the same as:
AlphaSymbolization Symbolization in EG Expression in PL P P ~
AlphaInference Rules Double Cut (De)Iteration Erasure 2k 1 2k 1 Insertion 2k+1 1 2k+1 1
AlphaMultiple Readings Possible Readings P Q P Q or Q P ~(P Q) or ~P ~Q P Q ~(P ~Q) or ~P Q or P Q P Q ~(~P ~Q) or P Q or ~P Q or ~(~Q ~P) or Q P or ~Q P P Q
(i) f() = T (ii) f([]) = F (iii) f(P) = P (iv) f([P]) = ~P (v) f([D]) = ~f(D) (vi) f(D1 D2) = f(D1) f(D2) (vii) f([D1 D2]) = f([D1]) f([D2]) (viii) f([D1 [D2 D3]]) = f(D1) f(D2) f([D1 [D3]]) (ix) f([[D]]) = f(D) (PROLOG Project: Given subset, generate all readings) AlphaMultiple Readings Formalized (see Shin)
AlphaRecursive Conditional Reading Q R S P This graph can be read as: P (Q (R S)), i.e. P Q & P (R S) or P Q & (P R) S
AlphaRelative Recursive Conditional Reading 0 1 2k-1 2k The Recursive Conditional Reading (RCR) relative to any subgraph 2k-1 or 2k is: 0 & 1 2 & (1 3 2k-1) 2k
AlphaSoundness of Insertion 0 & 1 2 & (1 3 2k-1) 2k 0 1 2k-1 2k Strengthening the Antecedent IN 0 & 1 2 & (1 3 2k-1 )2k 0 1 2k-1 2k
AlphaSoundness of Erasure 0 & 1 2 & (1 3 2k-1) (2k ) 0 1 2k-1 2k Weakening the Consequent E 0 & 1 2 & (1 3 2k-1)2k 0 1 2k-1 2k
AlphaSoundness of Iteration/Deiteration Case 1 & 1 2 & (1 3 2k-1) 2k 1 2k-1 2k p, q r p, q (r p) IT/DE & 1 2 & (1 3 2k-1)(2k ) 1 2k-1 2k
AlphaSoundness of Iteration/Deiteration Case 2 & 1 2 & (1 3 2k-1) 2k 1 2k-1 2k p, q r p, (q p) r IT/DE & 1 2 & (1 3 2k-1 )2k 1 2k-1 2k
